Precalculus Foundations: Absolute Value, Intervals, and Arithmetic Guidance

Q1. Rewrite the following expression without absolute value bars: |−345|

Background

Topic: Absolute Value

This question tests your understanding of absolute value, which measures the distance of a number from zero on the number line, regardless of direction.

Key Terms:

• Absolute value: $|a|$ is always non-negative.

Step-by-Step Guidance

1. Recall that the absolute value of a negative number is its positive counterpart.

2. Remove the absolute value bars and write the positive value of $-345$.

Try solving on your own before revealing the answer!

Final Answer: 345

$| -345 | = 345$

The absolute value of $-345$ is $345$ because absolute value always yields a non-negative result.

Q2. Rewrite the following expression without absolute value bars: |6 − 13|

Background

Topic: Absolute Value

This question tests your ability to evaluate the absolute value of an expression.

Key Terms:

• Absolute value: $|a|$

Step-by-Step Guidance

1. Calculate $6 - 13$.

2. Take the absolute value of the result.

Try solving on your own before revealing the answer!

Final Answer: 7

$|6 - 13| = |-7| = 7$

The absolute value of $-7$ is $7$.

Q3. Rewrite the expression without absolute value bars: |−14|

Background

Topic: Absolute Value

This question tests your understanding of absolute value for negative numbers.

Key Terms:

• Absolute value: $|a|$

Step-by-Step Guidance

1. Identify the value inside the absolute value bars.

2. Write the positive version of $-14$.

Try solving on your own before revealing the answer!

Final Answer: 14

$| -14 | = 14$

Absolute value turns negative numbers into their positive counterparts.

Q4. Rewrite the expression without absolute value bars: |−8 − (−14)|

Background

Topic: Absolute Value and Arithmetic

This question tests your ability to simplify expressions and apply absolute value.

Key Terms:

• Absolute value: $|a|$

• Arithmetic operations: subtraction and negative numbers

Step-by-Step Guidance

1. Calculate $-8 - (-14)$.

2. Take the absolute value of the result.

Try solving on your own before revealing the answer!

Final Answer: 6

$| -8 - (-14) | = | -8 + 14 | = |6| = 6$

Subtracting a negative is the same as adding its positive.

Q5. Graph the interval (−8, −2] on a number line and write the inequality notation.

Background

Topic: Intervals and Inequalities

This question tests your understanding of interval notation and how to represent it as an inequality.

Key Terms:

• Interval notation: $(a, b]$ means $a < x \leq b$

• Inequality notation: expresses the range of values for $x$

Step-by-Step Guidance

1. Identify the endpoints: $-8$ (not included), $-2$ (included).

2. Write the inequality that matches the interval.

Try solving on your own before revealing the answer!

Final Answer: $-8 < x \leq -2$

The interval $( -8, -2 ]$ corresponds to $-8 < x \leq -2$.

Q6. Graph the interval [7, ∞) on a number line and write the inequality notation.

Background

Topic: Intervals and Inequalities

This question tests your ability to convert interval notation to inequality notation.

Key Terms:

• Interval notation: $[a, \infty)$ means $x \geq a$

• Inequality notation: expresses the range of values for $x$

Step-by-Step Guidance

1. Identify the left endpoint: $7$ (included).

2. Write the inequality that matches the interval.

Try solving on your own before revealing the answer!

Final Answer: $x \geq 7$

The interval $[7, \infty)$ corresponds to $x \geq 7$.

Q7. Graph the interval (−∞, 1) on a number line and write the inequality notation.

Background

Topic: Intervals and Inequalities

This question tests your ability to interpret intervals that extend to infinity.

Key Terms:

• Interval notation: $(-\infty, b)$ means $x < b$

• Inequality notation: expresses the range of values for $x$

Step-by-Step Guidance

1. Identify the right endpoint: $1$ (not included).

2. Write the inequality that matches the interval.

Try solving on your own before revealing the answer!

Final Answer: $x < 1$

The interval $(-\infty, 1)$ corresponds to $x < 1$.

Q8. Graph the interval (1, 3) on a number line and write the inequality notation.

Background

Topic: Intervals and Inequalities

This question tests your ability to convert open intervals to inequalities.

Key Terms:

• Interval notation: $(a, b)$ means $a < x < b$

• Inequality notation: expresses the range of values for $x$

Step-by-Step Guidance

1. Identify both endpoints: $1$and $3$ (neither included).

2. Write the inequality that matches the interval.

Try solving on your own before revealing the answer!

Final Answer: $1 < x < 3$

The interval $(1, 3)$ corresponds to $1 < x < 3$.

Q9. Add and simplify: $\frac{13}{19} + \frac{3}{13}$

Background

Topic: Fraction Addition

This question tests your ability to add fractions with different denominators.

Key Terms and Formulas:

• Common denominator: $a/b + c/d = (ad + bc)/(bd)$

Step-by-Step Guidance

1. Find a common denominator for $19$and $13$.

2. Rewrite each fraction with the common denominator.

3. Add the numerators and keep the common denominator.

Try solving on your own before revealing the answer!

Final Answer: $\frac{226}{247}$

$\frac{13}{19} + \frac{3}{13} = \frac{(13 \times 13) + (3 \times 19)}{19 \times 13} = \frac{226}{247}$

We used the common denominator $247$ and combined the numerators.

Q10. Subtract and simplify: $\frac{13}{15} - \frac{2}{9}$

Background

Topic: Fraction Subtraction

This question tests your ability to subtract fractions with different denominators.

Key Terms and Formulas:

• Common denominator: $a/b - c/d = (ad - bc)/(bd)$

Step-by-Step Guidance

1. Find a common denominator for $15$and $9$.

2. Rewrite each fraction with the common denominator.

3. Subtract the numerators and keep the common denominator.

Try solving on your own before revealing the answer!

Final Answer: $\frac{29}{45}$

$\frac{13}{15} - \frac{2}{9} = \frac{(13 \times 9) - (2 \times 15)}{15 \times 9} = \frac{29}{45}$

We used the common denominator $45$ and subtracted the numerators.

Q11. Multiply and simplify: $\frac{10}{51} \cdot \frac{3}{8}$

Background

Topic: Fraction Multiplication

This question tests your ability to multiply fractions and simplify the result.

Key Terms and Formulas:

• Multiplication of fractions: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$

Step-by-Step Guidance

1. Multiply the numerators together.

2. Multiply the denominators together.

3. Simplify the resulting fraction if possible.

Try solving on your own before revealing the answer!

Final Answer: $\frac{5}{68}$

$\frac{10}{51} \cdot \frac{3}{8} = \frac{30}{408} = \frac{5}{68}$

We simplified the fraction by dividing numerator and denominator by $6$.

Q12. Compute and simplify: $\frac{4}{3} \div \frac{16}{9}$

Background

Topic: Fraction Division

This question tests your ability to divide fractions and simplify the result.

Key Terms and Formulas:

• Division of fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$

Step-by-Step Guidance

1. Rewrite the division as multiplication by the reciprocal.

2. Multiply the numerators and denominators.

3. Simplify the resulting fraction.

Try solving on your own before revealing the answer!

Final Answer: $\frac{3}{4}$

$\frac{4}{3} \div \frac{16}{9} = \frac{4}{3} \cdot \frac{9}{16} = \frac{36}{48} = \frac{3}{4}$

We simplified the fraction by dividing numerator and denominator by $12$.

Q13. Evaluate the expression for $x = 4$ and $y = -3$: $3(x + y) - 5y$

Background

Topic: Substitution and Simplification

This question tests your ability to substitute values into an expression and simplify.

Key Terms:

• Substitution: replacing variables with given values

• Simplification: combining like terms

Step-by-Step Guidance

1. Substitute $x = 4$ and $y = -3$ into the expression.

2. Calculate $x + y$.

3. Multiply the result by $3$.

4. Calculate $-5y$.

5. Add the two results together.

Try solving on your own before revealing the answer!

Final Answer: 18

$3(4 + (-3)) - 5(-3) = 3(1) + 15 = 18$

Substituting and simplifying gives the final value.

Q14. Evaluate the following expression for $x = 21$ and $y = -15$: $x - 9y + xy$

Background

Topic: Substitution and Simplification

This question tests your ability to substitute values and combine terms.

Key Terms:

• Substitution: replacing variables with given values

• Simplification: combining like terms

Step-by-Step Guidance

1. Substitute $x = 21$ and $y = -15$ into the expression.

2. Calculate $-9y$.

3. Calculate $xy$.

4. Add all terms together: $x$, $-9y$, and $xy$.

Try solving on your own before revealing the answer!

Final Answer: -237

$21 - 9(-15) + (21)(-15) = 21 + 135 - 315 = -237$

Substituting and simplifying gives the final value.